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A Unified Framework for Forward and Inverse Problems in Subsurface Imaging using Latent Space Translations
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Proposes a Generalized Forward-Inverse (GFI) framework based on manifold and latent-space-translation assumptions, instantiated as Latent U-Net and Invertible X-Net for jointly solving forward and inverse seismic subsurface imaging on OpenFWI.
Problem
Learning mappings between velocity maps and seismic waveforms in both directions (forward modeling and full waveform inversion) for subsurface imaging, with open questions on latent size, translation complexity, manifold learning, and joint forward/inverse training.
Approach
Unifies prior DL4SI methods as special cases of encoder-latent translation-decoder architectures; proposes Latent U-Net (two U-Nets for latent translation in each direction) and Invertible X-Net (a single invertible IU-Net translating latents both ways, trained jointly).
Key results
- Latent U-Net (Large) achieves the best inverse-problem average rank (1.3) on MAE across 10 OpenFWI datasets, outperforming InversionNet, Auto-Linear.
- Invertible X-Net achieves the best forward-problem average rank on SSIM (1.1) and MAE (1.4) on OpenFWI, with 26M parameters versus 35M for Latent U-Net.
- Zero-shot results on Marmousi and Overthrust show Latent U-Net and Invertible X-Net delineate sharp velocity changes better than baselines, though overall SSIM values remain low (around 0.44-0.50).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Latent U-Net (Large) instead of the baselines | Inverse modeling · OpenFWI | MAE: 1.3 average rank | Helped |
| Use Invertible X-Net instead of the baselines | Forward modeling · OpenFWI | SSIM: 1.1 average rank | Helped |
| Use smaller latent dimensions (ablation) | Latent U-Net (Large) and Invertible X-Net · OpenFWI A-family inverse | MAE | No clear effect |
| Use smaller latent dimensions (ablation) | Latent U-Net (Large) and Invertible X-Net · OpenFWI B-family inverse | MAE | Hurt |
| Use Latent U-Net (Large) instead of Latent U-Net (Small) (ablation) | Latent U-Net · OpenFWI inverse | translation performance | Helped |
| Remove skip connections from latent translation (ablation) | Latent U-Net · OpenFWI inverse with latent dimensions below 20 | translation performance | Hurt |
| Add skip connections to latent translation (ablation) | Latent U-Net · OpenFWI inverse with larger latent dimensions | translation performance | Helped |
| Use directly translating latent spaces instead of reconstruction followed by translation or combined reconstruction and translation (ablation) | Latent U-Net (Large) · OpenFWI | translation performance | Helped |
| Train Invertible X-Net on forward loss only instead of jointly on forward and inverse losses (ablation) | Invertible X-Net · OpenFWI forward | performance | Hurt |
| Jointly optimize forward and inverse losses (ablation) | Invertible X-Net · OpenFWI forward | performance | Helped |
| Add cycle loss (ablation) | Invertible X-Net · OpenFWI | generalization performance | Mixed |
Surprising findings
- Invertible X-Net trained only on the forward problem performs worse than the jointly trained model, and the joint training initially focuses on inversion before shifting to forward prediction.
- Manifold learning via reconstruct-then-translate underperforms direct latent translation training, suggesting latent spaces optimized solely for translation are most effective.
Limitations
- Real-world-like zero-shot performance remains weak (SSIM about 0.44-0.50 on Marmousi and Overthrust), and some baselines outperform the proposed models on several metrics.
- Invertibility holds only for the latent translation model; encoder-decoder pairs are not invertible, so the whole model is not invertible.
- Experiments are limited to 70x70 velocity maps and 2D OpenFWI-style data; cycle-loss unpaired training is not thoroughly evaluated in the main results.
Experimental setup
Datasets: OpenFWI (FlatVel-A/B, CurveVel-A/B, FlatFault-A/B, CurveFault-A/B, Style-A/B) Marmousi Overthrust
Benchmarks: OpenFWI Marmousi Overthrust
Metrics: MAE MSE SSIM
Baselines: InversionNet VelocityGAN Auto-Linear FNO WaveformNet
Builds on: InversionNet VelocityGAN Auto-Linear Fourier Neural Operator IU-Net Latent Space Mapping
Closest papers by meaning
- Unsupervised Learning of Full-Waveform Inversion: Connecting CNN and Partial Differential Equation in a Loop (ICLR 2022 · similarity 0.92)
- OpenFWI: Large-scale Multi-structural Benchmark Datasets for Full Waveform Inversion (NeurIPS 2022 · similarity 0.89)
- Auto-Linear Phenomenon in Subsurface Imaging (ICML 2024 · similarity 0.88)
- A Novel Diffusion Model for Pairwise Geoscience Data Generation with Unbalanced Training Dataset (AAAI 2025 · similarity 0.85)
- NPN: Non-Linear Projections of the Null-Space for Imaging Inverse Problems (NeurIPS 2025 · similarity 0.85)
- Integrating Intermediate Layer Optimization and Projected Gradient Descent for Solving Inverse Problems with Diffusion Models (ICML 2025 · similarity 0.84)
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